Abstract
Extension theorems are common in various areas of mathematics. In topology continuous extensions of continuous functions are studied. In functional analysis one is interested mainly in linear extensions of linear operators preserving continuity or some other properties like bounds or norm. In algebra extensions of homomorphisms and isomorphisms are investigated. The latter can be considered as extensions of functional equations.
Research supported by the Natural Sciences and Engineering Research Council of Canada grants nr. OGP002972 and CPG0164211
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York-London, 1966 [Mathematics in Science and Engineering, Vol. 19].
J. Aczél, Some good and bad characters I have known and where they led. (Harmonic analysis and functional equations), In: 1980 Seminar on Harmonic Analysis. [Canad. Math. Soc. Conf. Proc., Vol. 1]. Amer Math. Soc., Providence, RI, 1981, pp. 177–187.
J. Aczél, Diamonds are not the Cauchy extensionist’s best friend, C. R. Math. Rep. Acad. Sci. Canada 5 (1983), 259—264.
J. Aczél, it 28. Remark, Report of Meeting. The Twenty-second International Symposium on Functional Equations (December 16-December 22, 1984, Oberwolfach, Germany). Aequationes Math. 29 (1985), p. l01.
J. Aczél, J. A. Baker, D. Z. Djoković, P1. Kannappan and F. Radó, Extensions of certain homomorphisms of subsemigroups to homomorphisms of groups, Aequationes Math. 6 (1971), 263–271.
J. Aczél and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge-New York-New Rochelle-Melbourne-Sydney, 1989.
J. Aczél and P. Erdős, The non-existence of a Hamel-basis and the general solution of Cauchy’s functional equation for nonnegative numbers, Publ. Math. Debrecen 12 (1965), 259–265.
J. Aczél, P1. Kannappan, C. T. Ng and C. Wagner, Functional equations and inequalities in ‘rational group decision making’, In: General Inequalities 3 (Proc. Third Internat. Conf. on General Inequalities, Oberwolfach, 1981). Birkh auser, Basel-Boston-Stuttgart, 1983, pp. 239–243.
J. Aczél, C. T. Ng and C. Wagner, Aggregation theorems for allocation problems, SIAM J. Alg. Disc. Meth. 5 (1984), 1–8.
J. Aczél and C. Wagner, Rational group decision making generalized: the case of several unknown functions, C. R. Math. Rep. Acad. Sci. Canada 3 (1981), 139–142.
N. G. de Bruijn, On almost additive functions, Colloquium Math. 15 (1966), 59–63.
B. Crstici, I. Muntean and N. Vornicescu, General solution of the arctangent functional equation, Anal. Numér, Théor. Approx. 12 (1983), 113–123.
Z. Daróczy and L. Losonczi, Uber die Erweiterung der auf einer Punktmenge additiven Funktionen, Publ. Math. Debrecen 14 (1967), 239–245.
J. Dhombres and R. Ger, Conditional Cauchy equations, Glas. Mat. Ser. III. 13 (33) (1978), 39–62.
P. Erdős, P 310, Colloquium Math. 7 (1960), 311.
R. Ger, On almost polynomial functions, Colloquium Math. 24 (1971),95–101.
R. Ger, On some functional equations with a restricted domain, Fundamenta Math. 89 (1975), 95-l01.
S. Hartman, A remark about Cauchy’s equation, Colloquium Math. 8 (1961), 77–79.
W. B. Jurkat, On Cauchy’s functional equation, Proc. Amer. Math. Soc. 16 (1965), 683–686.
H. Kiesewetter, Uber die arc tan-Funktionalgleichung, ihre mehrdeutigen stetigen Losunqeti und eine nichtstetige Gruppe, Wiss. Z. Friedrich-Schiller-Univ. Jena Math.-Natur. 14 (1965), 417–421.
M. Kuczrna, Almost convex functions, Colloquium Math. 21 (1970), 279–284.
M. Kuczma, Functional equations on restricted domains, Aequationes Math. 18 (1978), 1–35.
K. Lajkó, Applications of extensions of additive functions, Aequationes Math. 11 (1974), 68–76.
L. Losonczi, An extension theorem, Aequationes Math. 28 (1985), 293–299.
L. Losonczi, Remark 32: The general solution of the arc tan equation, In: Proc. Twenty-third Internat. Symp. on Functional Equations (Gargnano, Italy, June 2–11, 1985). Univ. of Waterloo, Centre for Information Theory, Waterloo, Ont., 1985, pp. 74–76.
L. Losonczi, Local solutions of functional equations, Glasnik Mat. 25 (45) (1990), 57–67.
L. Losonczi, An extension theorem for the Levi-Cività. functional equation and its applications, Grazer Math. Ber. 315 (1991), 51–68.
S. C. Martin, Extensions and decompositions of homomorphisms of semigroups, Manuscript, University of Waterloo, Ont., 1977.
I. Muntean and N. Vornicescu, On the arctangent functional equation, (Roumanian), Seminarul “Theodor Angheluta”, Cluj-Napoca, 1983, pp. 241–246.
C. T. Ng, Representation for measures of information with the branching property, Inform. and Control 25 (1974), 45–56.
K. E. Osondu, Extensions of homomorphisms of a subsemigroup of a group, Semigroup Forum 15 (1978), 311–318.
L. Paganoni and J. R atz, Conditional functional equations and orthogonal additivity, Aequationes Math. 50 (1995), 134–141.
F. Radó and J. A. Baker, Pexider’s equation and aggregation of allocations, Aequationes Math. 32 (1987), 227–239.
J. Rimán, On an extension of Pexider’s equation, Zbornik Radova Mat. Inst. Beograd N. S. 1(9) (1976), 65–72.
L. Székelyhidi, The general representation of an additive function on an open point set, (Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. K ozl. 21 (1972), 503–509.
L. Székelyhidi, An extension theorem for a functional equation, Publ. Math. Debrecen 28 (1981), 275–279.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Aczél, J., Losonczi, L. (2013). Extension of Functional Equations. In: Graham, R., Nešetřil, J., Butler, S. (eds) The Mathematics of Paul Erdős I. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7258-2_28
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7258-2_28
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7257-5
Online ISBN: 978-1-4614-7258-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)